论文标题
凸约限制下的嘈杂线性反问题：高维
Noisy linear inverse problems under convex constraints: Exact risk asymptotics in high dimensions
论文作者
论文摘要
在标准的高斯线性测量模型中，$ y =xμ0+ξ\ in \ mathbb {r}^m $，带有固定噪声级别$σ> 0 $，我们考虑了估计未知信号$μ_0$在convex约束下$μ__0\ in k $ in $ k $ in $ k $的问题我们表明，自然凸的最小二乘估计量（LSE）$ \hatμ（σ）$的风险可以通过凸的约束的LSE $ \hatμ_k^{\ Mathsf {seq}} $在相应的高斯序列模型中以不同的噪声级别在不同的噪声模型中的凸面上的LSE $ \hatμ_k^{\ Mathsf {seq}} $。在最大制度中的风险（统一）的表征范围从恒定顺序一直到本质上是参数速率，只要满足$ \hatμ（σ）$的某些必要的非统一条件。 精确的风险表征揭示了无噪声（或低噪声限制）和噪音线性反向问题的基本差异，从而在信号恢复的样本复杂性方面。等渗回归问题给出了一个具体的示例：虽然一般单调信号的精确恢复需要$ m \ gg n^{1/3} $样品在无噪声设置中，但在嘈杂的设置中，一致的信号恢复需要少于$ m \ gg gg \ log \ log \ log n $ samples。当$ \hatμ_k^{\ mathsf {seq}} $的低噪声风险行为差异很大时，就会发生这种差异。在统计语言中，这发生在$ \hatμ_k^{\ mathsf {seq}} $估算$ 0 $的$ 0 $以比一般信号较慢的“最差案例率”时。还制定了其他几个示例，包括非阴性最小二乘和广义的套索（以约束形式），也可以证明该理论在不同类型的问题中的具体适用性。
In the standard Gaussian linear measurement model $Y=Xμ_0+ξ\in \mathbb{R}^m$ with a fixed noise level $σ>0$, we consider the problem of estimating the unknown signal $μ_0$ under a convex constraint $μ_0 \in K$, where $K$ is a closed convex set in $\mathbb{R}^n$. We show that the risk of the natural convex constrained least squares estimator (LSE) $\hatμ(σ)$ can be characterized exactly in high dimensional limits, by that of the convex constrained LSE $\hatμ_K^{\mathsf{seq}}$ in the corresponding Gaussian sequence model at a different noise level. The characterization holds (uniformly) for risks in the maximal regime that ranges from constant order all the way down to essentially the parametric rate, as long as certain necessary non-degeneracy condition is satisfied for $\hatμ(σ)$. The precise risk characterization reveals a fundamental difference between noiseless (or low noise limit) and noisy linear inverse problems in terms of the sample complexity for signal recovery. A concrete example is given by the isotonic regression problem: While exact recovery of a general monotone signal requires $m\gg n^{1/3}$ samples in the noiseless setting, consistent signal recovery in the noisy setting requires as few as $m\gg \log n$ samples. Such a discrepancy occurs when the low and high noise risk behavior of $\hatμ_K^{\mathsf{seq}}$ differ significantly. In statistical languages, this occurs when $\hatμ_K^{\mathsf{seq}}$ estimates $0$ at a faster `adaptation rate' than the slower `worst-case rate' for general signals. Several other examples, including non-negative least squares and generalized Lasso (in constrained forms), are also worked out to demonstrate the concrete applicability of the theory in problems of different types.
